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by: Dr. Simeon Wiza


Dr. Simeon Wiza
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This 7 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 122 at University of Washington taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/192445/phys-122-university-of-washington in Physics 2 at University of Washington.


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Date Created: 09/09/15
Lecture 12 Vector Analysis I See Chapter 6 in Boas Now we want to combine our earlier discussion of vectors with our more recent discussion of partial derivatives and functions of several variables Recall from Lecture 6 that we have de ned various products for vectors The rst is the scalar product 3 Exixz y1y2 2122 ElliM Zl llfilcosqz39 12391 k1 An example of how this is applied in physics is when calculating the work done by a force F on an object that moves through a path element at dW F 05 The second product is the vector or cross product that produces a vector orthogonal to both of the original vectors with a sense provided by the righthandrule ii Xiz yIZZ y221czlx2 sz13 x1y2 x2y12 3 a a 122 71 X 72 k Z gkbnrlJrzm E gkbn iJrzm 1 1 An example of how this quantity arises in physics is in the analysis a j of motion in a circle Consider a particle with position de ned by 7 moving in a circle of radius p rsinQ at constant angular velocity a with the origin of coordinates on the perpendicular through the center of the circle as indicated in the gure We associate this trajectory with a vector valued angular velocity E with magnitude 0 and vector direction orthogonal to the plane of the circle with sense give by the righthandrule applied to the motion about the circle Then we have the usual linear vector velocity given by 17 1 C X F which is tangent at each point to the circle describing the path of motion Vi A third product arises from combining the rst two This yields the triple scalar product de ned by 3 fisz HZEmma EXEE xfi 123 mm Physics 227 Lecture 12 1 Autumn 2007 This product will arise for example when we take the component of a torque r foalongavector f lt17x Something we did not mention in Lecture 6 is the triple vector product defined by F142XE EXEX 39EE 39 124 Remarkably we can obtain an understanding of the righthandside of this last equation by effectively pure thought We apply a very useful technique that can be called either the What else can it be theorem or the Only game in town theorem The basic idea is that we want to define a vector constructed out of the 3 vectors ii 72 and 173 which has the following general properties a the form of the vector can only know about the vector directions defined by 1713172 and 173 and cannot depend on any specific choice of basis vectors39 b due to the structure of the lefthandside of Eq 124 each term on the righthandside must be linear in each of the vectors39 c as a vector the righthandside must be orthogonal to a due to the left most cross product Thus pure thought says the most general allowed form is axzxacaafz M 025 where the constant c is independent of the specific vectors Each term has the required properties and we have arranged that the righthandside is orthogonal to f We can determine 0 by considering a specific example say 171 12172 12173 3 Explicit calculation yields 171 x x 3 x 2 f and cF1F3F2 F1 f2173c f2 so that 01 Such a product will arise in the case of circular motion when we evaluate the angular momentum LFxfyxm mfxa3xf Since things get interesting when we consider variation with time we need to develop the required notation for time derivatives of vectors something we have already used If we use a fixed set of basis vectors ie the basis vectors are not themselves functions of time fc 2 0 we have Physics 227 Lecture 12 2 Autumn 2007 Fx yf2zi 17imeyo22x z k i 126 dt dt dt dt i b z22 dzy d 222 all2 all2 all2 all2 We also have various versions of the chain rule For a product of a scalar function of time times a vector function of time we have 521 zagngg 127 Similarly we have the two forms a a 128 1gxg jxgax dt dt dt where in the first line the order of the factors in each term does not matter while in the second line the order matters ie there is a potential factor of l Once more we will illustrate some of these ideas by considering uniform motion in a circle With the geometry as defined in the earlier figure moving on a circle means I72 17 17 r02 pOZsin2 60 where r0 po 60 are constants Uniform motion means 172 72 2 V02 with V0 a scalar constant Taking time derivatives of these quantities we obtain the following properties of the various vectors 1 F2 1 r2 202 F217F317li all dt 0 r v 02vv2avgtalv 129 amlcgtlflr Physics 227 Lecture 12 3 Autumn 2007 In words we say that for uniform circular motion the acceleration is orthogonal to the velocity which is orthogonal to the position vector Taking one more derivative we have li k 0vVaiEa Him In the special case that we take the origin to be at the center of the circular orbit so that re is the radius of the circle we have the familiar result that the radial or centripetal acceleration is given by ar 2 v2r0 Circular motion is a natural setting for the use of curvilinear coordinates but it is important to recall that for such coordinates the unit basis vectors themselves are functions of the coordinates and will likely be time dependent The most useful curvilinear basis vectors are those for cylindrical coordinates and for spherical coordinates Compared to the rectilinear basis vectors the cylindrical basis vectors look like Z Z plx2y2 030300 x pcos lt3 I tan OS lt27 y psm f x f ww Basis Vectors 2 cos sin fcsin cosg 2 2 d azn 5 x g 2 plus cyclical permutations Path Length squared als2 dx2 aly2 dz2 gt dis2 2 61102 pzdg z dzz The corresponding spherical basis vectors are Physics 227 Lecture 12 4 Autumn 2007 1 2quotc2yzz2 OSrSOO z rcos6 2 2 ix x rs1n8cos Cgt 6tan 1 y 03837 2 y rsin8sin tan 1Z 0 lt27z x Basis Vectors 196 W sin8cos sin8sin f2 cos82 sin 65 cos 62 8 cos8cos c cos6sin f2 sin62 cos6 sin62 1212 csin f2cos z g z zo 5 X 3 19 plus cyclical permutations Path Length squared ds2 dx2 dy2 dz2 gt ds2 2 dr2 I 2d82 r2 sin2 8d 2 Note that the azimuthal angle is the same periodic angle in both cylindrical and spherical coordinate systems 0 3 lt 27 with the endpoints of the interval identified but that the spherical polar angle 6 varies only on the limited range 0 to 7 0 S 6 S 7 All of the various unit vector sets form righthanded orthogonal triplets As a first illustration we return to the familiar case of motion on a circle again with the geometry of the previous figure In cylindrical coordinates we place the circle in the xy plane at fixed 2 20 or in spherical coordinates at fixed 6 60 The simplest case is 20 0 60 7r2 For uniform motion we define t mot and identify Physics 227 Lecture 12 5 Autumn 2007 f6 60 t sin60b t cos602 sin 60 cos 5c sin j2 cos 602 f6 60 t sin60b t a sin6 0 sin 5c cos j2 wsin60 3t 1213 56 60 t cos603 t sin602 d A A A d A E6 cos60p wcos60 t E2 0 d A A A A E a cos x s1n y ap Finally as earlier we set the radius of the circle to a constant f6 60 r0f6 60 r0 sin603cos60r02 p03202 1214 and find the motion to be described by the time dependence of the 6 unit basis vector ie r0 60 po 20 and 2 are all constant in time V 76 90 p03 pow IV i 660p05p0w p0w23 1215 This last result is just the familiar centripetal acceleration directed radially inward towards the aXis of the circular motion Note that as advertised in Eq 129 the location vector 17 0C 9 Sin G O COS 902 is orthogonal to the velocity vector 7 0C amp which is orthogonal to the acceleration vector 7 0C 35 but is not necessarily orthogonal to Now we want to combine the earlier discussions and introduce a notation for partial derivatives that also carries information about the vector direction of the individual Physics 227 Lecture 12 6 Autumn 2007


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